Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications
نویسندگان
چکیده
The Singular Value Decomposition (SVD) of a matrix is a linear algebra tool that has been successfully applied to a wide variety of domains. The present paper is concerned with the problem of estimating the Jacobian of the SVD components of a matrix with respect to the matrix itself. An exact analytic technique is developed that facilitates the estimation of the Jacobian using calculations based on simple linear algebra. Knowledge of the Jacobian of the SVD is very useful in certain applications involving multivariate regression or the computation of the uncertainty related to estimates obtained through the SVD. The usefulness and generality of the proposed technique is demonstrated by applying it to the estimation of the uncertainty for three different vision problems, namely self-calibration, epipole computation and rigid motion estimation. Key-words: Singular Value Decomposition, Jacobian, Uncertainty, Calibration, Structure from Motion. M. Lourakis was supported by the VIRGO research network (EC Contract No ERBFMRX-CT96-0049) of the TMR Programme. Calcul de la Jacobienne de la Décomposition en Valeurs Singulières: Théorie et applications Résumé : La technique de Décomposition en Valeurs Singulières (SVD) d’une matrice est un outil algèbrique qui a trouvé de nombreuses applications en vision par ordinateur. Dans ce rapport, nous nous intéressons au problème de l’estimation de la jacobienne de la SVD par rapport aux coefficients de la matrice initiale. Cette jacobienne est très utile pour toute une gamme d’applications faisant intervenir des estimations aux moindres carrés (pour lesquelles on utilise la SVD) ou bien des calculs d’incertitude pour des grandeurs estimées de cette manière. Une solution analytique simple à ce problème est présentée. Elle exprime la jacobienne à partir de la SVD de la matrice à l’aide d’opérations très simples d’algèbre linéaire. L’utilité et la généralité de la technique est démontrée en l’appliquant à trois problèmes de vision: l’auto-calibration, le calcul d’épipoles et l’estimation de mouvements rigides. Mots-clés : Décomposition en valeurs singulières, Jacobienne, Incertitude, Calibration, Structure à partir du mouvement. Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications 3
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